Question

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=–5$, then the distance between $A$ and $B$ is

Answer 1

Let $P(t^2,2t)$ be a point on the parabola $y^2=4x$ and $Q(h,k)$ be its mirror image with respect to the line $x+y+4=0.$
$\text{Then}\frac{h-t^2}{1}=\frac{k-2t}{1}=-2\left(\frac{t^2+2t+4}{1^2+1^2}\right)\Rightarrow h-t^2=-(t^2+2t+4)\&k-2t=-(t^2+2t+4)\Rightarrow h=-2t-4\&k=-t^2-4\text{Now,}k=–5\Rightarrow t=\pm 1\Rightarrow h=–2,–6\Rightarrow A,B\text{are}(–2,–5)\&(–6,–5)\text{respectively}\therefore AB=4$